Abstract

A method is developed for interpreting the statistics of the sun’s glitter on the sea surface in terms of the statistics of the slope distribution. The method consists of two principal phases: (1) of identifying, from geometric considerations, any point on the surface with the particular slope required for the reflection of the” sun’s rays toward the observer; and (2) of interpreting the average brightness of the sea surface in the vicinity of this point in terms of the frequency with which this particular slope occurs. The computation of the probability of large (and infrequent) slopes is limited by the disappearance of the glitter into a background consisting of (1) the sunlight scattered from particles beneath the sea surface, and (2) the skylight reflected by the sea surface.

The method has been applied to aerial photographs taken under carefully chosen conditions in the Hawaiian area. Winds were measured from a vessel at the time and place of the aerial photographs, and cover a range from 1 to 14 m sec−1. The effect of surface slicks, laid by the vessel, are included in the study. A two-dimensional Gram-Charlier series is fitted to the data. As a first approximation the distribution is Gaussian and isotropic with respect to direction. The mean square slope (regardless of direction) increases linearly with the wind speed, reaching a value of (tan16°)2 for a wind speed of 14 m sec−1. The ratio of the up/ downwind to the crosswind component of mean square slope varies from 1.0 to 1.9. There is some up/downwind skewness which increases with increasing wind speed. As a result the most probable slope at high winds is not zero but a few degrees, with the azimuth of ascent pointing downwind. The measured peakedness which is barely above the limit of observational error, is such as to make the probability of very large and very small slopes greater than Gaussian. The effect of oil slicks covering an area of one-quarter square mile is to reduce the mean square slopes by a factor of two or three, to eliminate skewness, but to leave peakedness unchanged.

© 1954 Optical Society of America

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References

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  1. J. Spooner, Corresp. Astro, du Baron de Zach, 6 (1822).
  2. E. O. Hulbert, J. Opt. Soc. Am. 24, 35 (1934).
    [Crossref]
  3. V. V. Shuleikin, Fizika Moria (Physics of the Sea) (Izdatelstvo Akad. Nauk. U.S.S.R., Moscow, 1941).
  4. M. Minnaert, Physica 9, 925 (1942).
    [Crossref]
  5. J. S. Van Wieringen, Proc. Koninkl. Ned. Akad. Wetenschap. 50, 952 (1947).
  6. Carl Eckart, The sea surface and its effect on the reflection of sound and light. University of Calif. Div. of War Research No. M407, March20, 1946 (unpublished). The horizontal extent of the glitter pattern is assumed small compared to the distance from the observer.
  7. C. Cox and W. H. Munk, Scripps Inst. of Oceanogr. Bull. (to be published).
  8. C. Eckart, J. Acoust. Soc. Am. 25, 566 (1953). Here the linearized equations are developed for the case where the incoming radiation is not necessarily short compared to the ocean waves. For the limiting case of short-wave radiation, Eckart obtains essentially p= 4ρ−1(N cosμ/H).
    [Crossref]
  9. For example, Harald Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946). This gives a general account, but the development in two dimensions could not be found in the literature. Details are given in reference 7.
  10. This accounts for the fact that one rarely sees distant trees, dunes, or ships reflected in the sea. The reason, as pointed out by Minnaert [M. Minnaert, Light and Colour in the Upper Air (G. Bell and Sons, London, 1940)], is that “at a great distance one sees only the sides of waves turned toward us. This makes it seem as if we saw all the objects … reflected in a slanting mirror.” For the same reason, the reflection of low, distant clouds is displaced toward the horizon.
  11. S. Q. Duntley, “The visibility of submerged objects.” Part I, Optical Effects of Water Waves. Mass. Inst. Tech. Report, Dec.15, 1950, on U. S. Office of Naval Research.
  12. A. H. Schooley, J. Opt. Soc. Am. 44, 37 (1954).
    [Crossref]

1954 (1)

1953 (1)

C. Eckart, J. Acoust. Soc. Am. 25, 566 (1953). Here the linearized equations are developed for the case where the incoming radiation is not necessarily short compared to the ocean waves. For the limiting case of short-wave radiation, Eckart obtains essentially p= 4ρ−1(N cosμ/H).
[Crossref]

1947 (1)

J. S. Van Wieringen, Proc. Koninkl. Ned. Akad. Wetenschap. 50, 952 (1947).

1942 (1)

M. Minnaert, Physica 9, 925 (1942).
[Crossref]

1934 (1)

1822 (1)

J. Spooner, Corresp. Astro, du Baron de Zach, 6 (1822).

Cox, C.

C. Cox and W. H. Munk, Scripps Inst. of Oceanogr. Bull. (to be published).

Cramer, Harald

For example, Harald Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946). This gives a general account, but the development in two dimensions could not be found in the literature. Details are given in reference 7.

Duntley, S. Q.

S. Q. Duntley, “The visibility of submerged objects.” Part I, Optical Effects of Water Waves. Mass. Inst. Tech. Report, Dec.15, 1950, on U. S. Office of Naval Research.

Eckart, C.

C. Eckart, J. Acoust. Soc. Am. 25, 566 (1953). Here the linearized equations are developed for the case where the incoming radiation is not necessarily short compared to the ocean waves. For the limiting case of short-wave radiation, Eckart obtains essentially p= 4ρ−1(N cosμ/H).
[Crossref]

Eckart, Carl

Carl Eckart, The sea surface and its effect on the reflection of sound and light. University of Calif. Div. of War Research No. M407, March20, 1946 (unpublished). The horizontal extent of the glitter pattern is assumed small compared to the distance from the observer.

Hulbert, E. O.

Minnaert, M.

M. Minnaert, Physica 9, 925 (1942).
[Crossref]

This accounts for the fact that one rarely sees distant trees, dunes, or ships reflected in the sea. The reason, as pointed out by Minnaert [M. Minnaert, Light and Colour in the Upper Air (G. Bell and Sons, London, 1940)], is that “at a great distance one sees only the sides of waves turned toward us. This makes it seem as if we saw all the objects … reflected in a slanting mirror.” For the same reason, the reflection of low, distant clouds is displaced toward the horizon.

Munk, W. H.

C. Cox and W. H. Munk, Scripps Inst. of Oceanogr. Bull. (to be published).

Schooley, A. H.

Shuleikin, V. V.

V. V. Shuleikin, Fizika Moria (Physics of the Sea) (Izdatelstvo Akad. Nauk. U.S.S.R., Moscow, 1941).

Spooner, J.

J. Spooner, Corresp. Astro, du Baron de Zach, 6 (1822).

Van Wieringen, J. S.

J. S. Van Wieringen, Proc. Koninkl. Ned. Akad. Wetenschap. 50, 952 (1947).

Corresp. Astro, du Baron de Zach (1)

J. Spooner, Corresp. Astro, du Baron de Zach, 6 (1822).

J. Acoust. Soc. Am. (1)

C. Eckart, J. Acoust. Soc. Am. 25, 566 (1953). Here the linearized equations are developed for the case where the incoming radiation is not necessarily short compared to the ocean waves. For the limiting case of short-wave radiation, Eckart obtains essentially p= 4ρ−1(N cosμ/H).
[Crossref]

J. Opt. Soc. Am. (2)

Physica (1)

M. Minnaert, Physica 9, 925 (1942).
[Crossref]

Proc. Koninkl. Ned. Akad. Wetenschap. (1)

J. S. Van Wieringen, Proc. Koninkl. Ned. Akad. Wetenschap. 50, 952 (1947).

Other (6)

Carl Eckart, The sea surface and its effect on the reflection of sound and light. University of Calif. Div. of War Research No. M407, March20, 1946 (unpublished). The horizontal extent of the glitter pattern is assumed small compared to the distance from the observer.

C. Cox and W. H. Munk, Scripps Inst. of Oceanogr. Bull. (to be published).

For example, Harald Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946). This gives a general account, but the development in two dimensions could not be found in the literature. Details are given in reference 7.

This accounts for the fact that one rarely sees distant trees, dunes, or ships reflected in the sea. The reason, as pointed out by Minnaert [M. Minnaert, Light and Colour in the Upper Air (G. Bell and Sons, London, 1940)], is that “at a great distance one sees only the sides of waves turned toward us. This makes it seem as if we saw all the objects … reflected in a slanting mirror.” For the same reason, the reflection of low, distant clouds is displaced toward the horizon.

S. Q. Duntley, “The visibility of submerged objects.” Part I, Optical Effects of Water Waves. Mass. Inst. Tech. Report, Dec.15, 1950, on U. S. Office of Naval Research.

V. V. Shuleikin, Fizika Moria (Physics of the Sea) (Izdatelstvo Akad. Nauk. U.S.S.R., Moscow, 1941).

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Figures (8)

F. 1
F. 1

Glitter patterns photographed by aerial camera pointing vertically downward at solar elevation of ϕ = 70°. The superimposed grids consist of lines of constant slope azimuth α (radial) drawn for every 30°, and of constant tilt β (closed) for every 5°. Grids have been translated and rotated to allow for roll, pitch, and yaw of plane. Shadow of plane can barely be seen along α = 180° within white cross. White arrow shows wind direction. Left: water surface covered by natural slick, wind 1.8 m sec−1, rms tilt σ = 0.0022. Right: clean surface, wind 8.6 m sec−1, σ = 0.045. The vessel Reverie is within white circle.

F. 2
F. 2

The coordinate system is centered at the sea surface “with the z-axis vertically upward (not shown) and the y-axis drawn horizontally toward the sun. The incident ray is reflected at A and forms an image at P on a horizontal photographic plate. Points A, B, C, and D define a horizontal plane through A and AB′C′D′, the plane tangent to the sea surface. The tilt β is measured in the direction AC of steepest ascent, and this direction makes an angle α to the right of the sun. OO′ is parallel to the z-axis and O′Y′ to the (negative) y-axis.

F. 3
F. 3

The tolerance ellipse.

F. 4
F. 4

The radiance (in arbitrary units) of a smooth sea surface (σ =0) and a rough sea surface (σ = 0.2), according to Eq. (16). The two branches of the latter curve for large μ correspond to two extreme assumptions regarding multiple reflections.

F. 5
F. 5

The solid circles show the measured background on a very calm day (3 Sept j, σ = 0.091), when the glitter pattern was confined to a small portion of the photograph. This background is due partly to reflected skylight, partly scattered sunlight, as shown.

F. 6
F. 6

The background radiation (in arbitrary units) on a moderately rough day (4 Sept k, σ = 0.13). The solid circles correspond to measurements outside the glitter pattern. The open circles contain some glitter radiation, and lie, therefore, above the background curve. Circles with the vertical line correspond to measurements on the vertical photograph, the other circles to measurements on tilted photograph.

F. 7
F. 7

Logarithm of unnormalized probability p as a function of azimuth α relative to sun, for indicated values of slope angle β. Open circles indicate data from tilted camera; barred circles, vertical camera. All measurements from photographs 4 Sept. k. Curves are drawn according to the Gram Charlier representation, Eq. (18). The position of the principal (y′) axis is indicated relative to recorded wind direction.

F. 8
F. 8

Principal sections through the probability distribution surface p(zx′,zy). The upper curves are along the crosswind axis x′; the lower curves along the upwind axis y′. The solid curves refer to the observed distribution, the dashed to a Gaussian distribution of equal mean square slope components. The thin vertical lines show the scale for the standardized slope components ξ = zx′/σc and η = zy′/σu. The heavy vertical segments show the corresponding tilts β = 5°, 10°, ⋯, 25° for a wind speed of 10 m sec−1; the skewness shown in the lower curve is computed for this wind speed. The modes are marked by arrows.

Tables (1)

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Table I

Equations (36)

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z x = z / x = sin α tan β , z y = z / y = cos α tan β ,
a n = sin α sin β , b n = cos α sin β , c n = cos β .
a i = 0 , b i = cos ϕ , c i = sin ϕ ,
a r = sin ν sin μ , b r = cos ν sin μ , c r = cos μ .
a r a i = 2 a n cos ω , b r b i = 2 b n cos ω , c r c i = 2 c n cos ω ,
cos ω = cos β sin ϕ cos α sin β cos ϕ .
cos μ = 2 cos β cos ω sin ϕ , cot ν = cot α 1 2 csc α csc β sec ω cos ϕ .
Δ t = 1 4 π 2 sec 3 β 0 sec ω 0 ,
P ( z x 0 , z y 0 ) = p ( z x 0 , z y 0 ) Δ t ( z x 0 , z y 0 ) .
J = ρ ( ω ) Δ h H ( π 2 ) 1 sec β cos ω
N cos μ = P J ,
p = 4 ρ 1 ( cos 4 β ) N cos μ / H .
2 ρ ( ω ) = sin 2 ( ω ω ) csc 2 ( ω + ω ) + tan 2 ( ω ω ) cot 2 ( ω ω ) ,
E = 1 4 π d 2 s 2 τ ( λ ) t Δ u cos λ cos μ
N = C sec 4 λ ( τ ) 1 E ( D ) , C = ( 4 f 2 ) / ( π d 2 t ) .
N = sec μ N s ρ ( ω ) cos ω sec β p ( z x , z y ) d z x d z y ,
cos ω = cos β ( cos μ + z y sin μ ) .
N = ( π σ 2 ) 1 N s ρ ( μ ) ( 1 + a z y + b z y 2 + c z x 2 + ) × exp [ ( z x 2 + z y 2 ) / σ 2 ] d z x d z y ,
a = ( F / F ) , b = 1 2 + 1 2 ( F / F ) , c = 1 2 + 1 2 ( F / F ) cot μ .
k = σ 1 cot μ .
N = N s ρ ( μ ) { 1 2 [ 1 + I ( k ) ] + 1 2 π 1 2 a σ e k 2 + 1 4 b σ 2 [ 1 + I ( k ) 2 π 1 2 k e k 2 ] + 1 4 c σ 2 [ 1 + I ( k ) ] + } ,
I ( k ) = 2 π 1 2 0 k e t 2 d t
k = ( 2 σ ) 1 cot μ
p = [ two dimensional Gaussian distribution ] × [ 1 + i , j = 1 c i j H i ( ξ ) H j ( η ) ] .
ξ = z x / σ c , η = z y / σ u ,
p ( z x , z y ) = ( 2 π σ c σ u ) 1 exp 1 2 ( ξ 2 + η 2 ) × [ 1 1 2 c 21 ( ξ 2 1 ) η 1 6 c 03 ( η 3 3 η ) + ( 1 / 24 ) c 40 ( ξ 4 6 ξ 2 + 3 ) + 1 4 c 22 ( ξ 2 1 ) ( η 2 1 ) + ( 1 / 24 ) c 04 ( η 4 6 η 2 + 3 ) + ] .
σ c 2 = 0.003 + 1.92 × 10 3 W ± 0.002 r = 0.956 σ u 2 = 0.000 + 3.16 × 10 3 W ± 0.004 r = 0.945 σ c 2 + σ u 2 = 0.003 + 5.12 × 10 3 W ± 0.004 r = 0.986
σ c 2 = 0.003 + 1.84 × 10 3 W ± 0.002 r = 0.78 σ u 2 = 0.005 + 0.78 × 10 3 W ± 0.002 r = 0.70 σ c 2 + σ u 2 = 0.008 + 1.56 × 10 3 W ± 0.004 r = 0.77
c 21 = 0.01 0.0086 W ± 0.03
c 03 = 0.04 0.033 W ± 0.12 .
c 21 = 0.00 ± 0.02 and c 03 = 0.02 ± 0.05 .
c 40 = 0.40 ± 0.23 c 22 = 0.12 ± 0.06 c 04 = 0.23 ± 0.41
0.36 ± 0.24 0.10 ± 0.05 0.26 ± 0.31
p d z x d z y , m p d α d m , and p tan β sec 2 β d α d β
z x + 1 2 d z x , z y ± 1 2 d z y ; α ± 1 2 d α , m ± 1 2 d m ;
α ± 1 2 d α , β ± 1 2 d β ,